This article is a part of a project investigating the relationship between the dynamics of comple... more This article is a part of a project investigating the relationship between the dynamics of completely integrable or "close" to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for the Laplace-Beltrami operators ∆ t , t ∈ [0, 1], with Dirichlet, Neumann or Robin boundary conditions, associated with C 1 families of billiard tables (X, g t). We introduce a notion of weak
We prove that there exist potentials so that near them the focusing non-linear Schrödinger equati... more We prove that there exist potentials so that near them the focusing non-linear Schrödinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.
Nonlinear Analysis-theory Methods & Applications, Mar 1, 2022
In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in ... more In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H s 0 (T, R) for any s > −1/2 where H s 0 (T, R) denotes the subspace of the Sobolev space H s (T, R) of elements with mean 0. As an application we show that for any −1/2 < s < 0, the flow map of the Benjamin-Ono equation S t 0 : H s 0 (T, R) → H s 0 (T, R) is nowhere locally uniformly continuous in a neighborhood of zero in H s 0 (T, R).
We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space ... more We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space Hs(T,R) for any s ≥ 0. To this end we show that Tao’s gauge transform is a high frequency approximation of the nonlinear Fourier transform Φ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.
We give a natural geometric condition called geodesic compatibility that implies the existence of... more We give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a (pseudo)Riemannian metric. We prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition. We apply our results for obtaining an infinite family (hierarchy) of completely integrable flows on the complex projective plane CPn.
In this paper we prove that the Euler equation describing the motion of an ideal fluid in R d is ... more In this paper we prove that the Euler equation describing the motion of an ideal fluid in R d is well-posed in a class of functions allowing spatial asymptotic expansions as |x| → ∞ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with rapidly decaying initial data develop non-trivial spatial asymptotic expansions of the type considered here.
In this paper we prove approximation properties of the solutions of the defocusing NLS equation o... more In this paper we prove approximation properties of the solutions of the defocusing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces Hs with s1 remain bounded for all times.
We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and in certain... more We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and in certain spaces of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle $\pi/2$ with polynomial growth as $t\to\infty$. We apply these results to nonlinear heat equations on $\R^d$, including global existence in time.
We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spa... more We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of t, do not involve (spacial) logarithmic terms, and develop even when the initial data has fast decay at infinity. We discuss the evolution in time of the asymptotic terms and their approximation properties.
We develop a framework for studying quasi-periodic maps and diffeomorphisms on R n. As an applica... more We develop a framework for studying quasi-periodic maps and diffeomorphisms on R n. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on R n. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.
We prove that the heat equation on R d is well-posed in certain spaces of functions allowing spat... more We prove that the heat equation on R d is well-posed in certain spaces of functions allowing spatial asymptotic expansions as |x| → ∞ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle π/2 with polynomial growth as t → ∞. Generically, a large class of nonlinear heat flows have equilibrium solutions with spatial asymptotics of the considered type. We provide a simple nonlinear model that features global in time existence with such asymptotics at spatial infinity.
We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at i... more We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the approach of V. Arnold [1].
We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on R d wi... more We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on R d with initial data in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as |x| with β < 1/2. In particular, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. We identify the evolution space for initial data in the Schwartz class with a certain space of symbols.
We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie ... more We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.
In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operato... more In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.
We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spa... more We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of t, do not involve (spacial) logarithmic terms, and develop even when the initial data has fast decay at infinity. We discuss the evolution in time of the asymptotic terms and their approximation properties.
This article is a part of a project investigating the relationship between the dynamics of comple... more This article is a part of a project investigating the relationship between the dynamics of completely integrable or "close" to completely integrable billiard tables, the integral geometry on them, and the spectrum of the corresponding Laplace-Beltrami operators. It is concerned with new isospectral invariants and with the spectral rigidity problem for the Laplace-Beltrami operators ∆ t , t ∈ [0, 1], with Dirichlet, Neumann or Robin boundary conditions, associated with C 1 families of billiard tables (X, g t). We introduce a notion of weak
We prove that there exist potentials so that near them the focusing non-linear Schrödinger equati... more We prove that there exist potentials so that near them the focusing non-linear Schrödinger equation does not admit local Birkhoff coordinates. The proof is based on the construction of a local normal form of the linearization of the equation at such potentials.
Nonlinear Analysis-theory Methods & Applications, Mar 1, 2022
In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in ... more In this paper we prove that the Benjamin-Ono equation admits an analytic Birkhoff normal form in an open neighborhood of zero in H s 0 (T, R) for any s > −1/2 where H s 0 (T, R) denotes the subspace of the Sobolev space H s (T, R) of elements with mean 0. As an application we show that for any −1/2 < s < 0, the flow map of the Benjamin-Ono equation S t 0 : H s 0 (T, R) → H s 0 (T, R) is nowhere locally uniformly continuous in a neighborhood of zero in H s 0 (T, R).
We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space ... more We prove smoothing properties of the solutions of the Benjamin-Ono equation in the Sobolev space Hs(T,R) for any s ≥ 0. To this end we show that Tao’s gauge transform is a high frequency approximation of the nonlinear Fourier transform Φ for the Benjamin-Ono equation, constructed in our previous work. The results of this paper are manifestations of the quasi-linear character of the Benjamin-Ono equation.
We give a natural geometric condition called geodesic compatibility that implies the existence of... more We give a natural geometric condition called geodesic compatibility that implies the existence of integrals in involution of the geodesic flow of a (pseudo)Riemannian metric. We prove that if two metrics satisfy the condition of geodesic compatibility then we can produce a hierarchy of metrics that also satisfy this condition. We apply our results for obtaining an infinite family (hierarchy) of completely integrable flows on the complex projective plane CPn.
In this paper we prove that the Euler equation describing the motion of an ideal fluid in R d is ... more In this paper we prove that the Euler equation describing the motion of an ideal fluid in R d is well-posed in a class of functions allowing spatial asymptotic expansions as |x| → ∞ of any a priori given order. These asymptotic expansions can involve log terms and lead to a family of conservation laws. Typically, the solutions of the Euler equation with rapidly decaying initial data develop non-trivial spatial asymptotic expansions of the type considered here.
In this paper we prove approximation properties of the solutions of the defocusing NLS equation o... more In this paper we prove approximation properties of the solutions of the defocusing NLS equation on the circle by nearly linear flows. In addition we show that spatially periodic solutions of the defocusing NLS equation evolving in fractional Sobolev spaces Hs with s1 remain bounded for all times.
We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and in certain... more We prove that the heat equation on $\R^d$ is well-posed in weighted Sobolev spaces and in certain spaces of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle $\pi/2$ with polynomial growth as $t\to\infty$. We apply these results to nonlinear heat equations on $\R^d$, including global existence in time.
We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spa... more We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of t, do not involve (spacial) logarithmic terms, and develop even when the initial data has fast decay at infinity. We discuss the evolution in time of the asymptotic terms and their approximation properties.
We develop a framework for studying quasi-periodic maps and diffeomorphisms on R n. As an applica... more We develop a framework for studying quasi-periodic maps and diffeomorphisms on R n. As an application, we prove that the Euler equation is locally well posed in a space of quasi-periodic vector fields on R n. In particular, the equation preserves the spatial quasi-periodicity of the initial data. Several results on the analytic dependence of solutions on the time and the initial data are proved.
We prove that the heat equation on R d is well-posed in certain spaces of functions allowing spat... more We prove that the heat equation on R d is well-posed in certain spaces of functions allowing spatial asymptotic expansions as |x| → ∞ of any a priori given order. In fact, we show that the Laplacian on such function spaces generates an analytic semigroup of angle π/2 with polynomial growth as t → ∞. Generically, a large class of nonlinear heat flows have equilibrium solutions with spatial asymptotics of the considered type. We provide a simple nonlinear model that features global in time existence with such asymptotics at spatial infinity.
We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at i... more We consider classes of diffeomorphisms of Euclidean space with partial asymptotic expansions at infinity; the remainder term lies in a weighted Sobolev space whose properties at infinity fit with the desired application. We show that two such classes of asymptotic diffeomorphisms form topological groups under composition. As such, they can be used in the study of fluid dynamics according to the approach of V. Arnold [1].
We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on R d wi... more We study the well-posedness and the spatial behavior at infinity of perfect fluid flows on R d with initial data in a scale of weighted Sobolev spaces that allow spatial growth/decay at infinity as |x| with β < 1/2. In particular, we show that the solution of the Euler equation generically develops an asymptotic expansion at infinity with non-vanishing asymptotic terms that depend analytically on time and the initial data. We identify the evolution space for initial data in the Schwartz class with a certain space of symbols.
We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie ... more We define the group of almost periodic diffeomorphisms on $\mathbb{R}^n$ and on an arbitrary Lie group. We then study the properties of its Riemannian and Lie group exponential maps and provide applications to fluid equations. In particular, we show that there exists a geodesic of a weak Riemannian metric on the group of almost periodic diffeomorphisms of the line that consists entirely of conjugate points.
In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operato... more In this paper we provide new asymptotic estimates of the Floquet exponents of Schrödinger operators on the circle. By the same techniques, known asymptotic estimates of various others spectral quantities are improved.
We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spa... more We prove that the 2d Euler equation is globally well-posed in a space of vector fields having spatial asymptotic expansion at infinity of any a priori given order. The asymptotic coefficients of the solutions are holomorphic functions of t, do not involve (spacial) logarithmic terms, and develop even when the initial data has fast decay at infinity. We discuss the evolution in time of the asymptotic terms and their approximation properties.
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